please help, im messing up somewhere and i cannot find my mistake so please show your steps.
you are to design an open top box with a square bottom using no more than 1200 square inches of cardboard. what dimensions will maximize the volume?
you are to design an open top box with a square bottom using no more than 1200 square inches of cardboard. what dimensions will maximize the volume?
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x = base length, y = height
open box
A = x² + 4xy
1200 = x² + 4xy
y = 300/x - x/4
V = base area x height
V = x²y
V = x² (300/x - x/4)
V = 300x - x³/4
V' = 300 - ¾ x²
300 - ¾ x² = 0
x = 20
2nd derivative test for maximum V'' < 0
V'' = - 3x/2
V''(20) = -30
y = 300/x - x/4
y = 300/20- 20/4 = 15 - 5 = 10
dimensions:
square base: 20 inches
height: 10 inches
open box
A = x² + 4xy
1200 = x² + 4xy
y = 300/x - x/4
V = base area x height
V = x²y
V = x² (300/x - x/4)
V = 300x - x³/4
V' = 300 - ¾ x²
300 - ¾ x² = 0
x = 20
2nd derivative test for maximum V'' < 0
V'' = - 3x/2
V''(20) = -30
y = 300/x - x/4
y = 300/20- 20/4 = 15 - 5 = 10
dimensions:
square base: 20 inches
height: 10 inches